Answer
$Area = 8 \int_0^{\frac{\pi}{2}} (sin^2 \theta) d\theta$
Work Step by Step
1. Find the limits of integration:
The shaded region starts at $\theta = 0$, and ends at $\theta = \frac{\pi}{2}$;
2. Apply the formula for the area:
$Area = \frac{1}{2} \int_\alpha^\beta(f(\theta)^2)d\theta$
$Area = \frac{1}{2} \int_0^{\frac{\pi}{2}}(4 sin\theta)^2d\theta$
$Area = \frac{1}{2} \int_0^{\frac{\pi}{2}}(16 sin^2\theta)d\theta$
$Area = 16 \times \frac{1}{2} \int_0^{\frac{\pi}{2}}( sin^2\theta)d\theta$
$Area = 8 \int_0^{\frac{\pi}{2}}( sin^2\theta)d\theta$
